Let $M$ be a commutative monoid with zero. Then the condition $M^* = M \setminus \{0\}$ is very similar to the condition for a commutative ring to be a field. This analogy is also used in the work "Schemes over $\mathbb{F}_1$ and Zeta functions" by Connes and Consani. However they don't give these monoids a name.

A very silly idea might be to call them "monoid fields".

Question. How are these monoids called in the literature? If there is no existing terminology yet, which one would you propose?

The answer by BS tells us that in the non-commutative case these are called groups with zero. My question deals with the commutative case. I would like to have a proper name, not just a combination such as "abelian group with zero" (which is confusing anyway).

It would help to elaborate on the condition M*. For all I know, it looks like a submonoid with (a different) zero. Gerhard "Ask Me About System Design" Paseman, 2012.02.09
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Gerhard PasemanFeb 9 '12 at 17:44

Thanks! This seems to be an established terminology, you can also find it in other places. But as you might guess this does not really satisfy me. Since I'm interested in commutative monoids with zero, I would have to call them "commutative groups with zero" or "abelian groups with zero". But this is confusing since abelian groups are usually written additively and the identity element is written as a zero. Also I would like to use a short name (similar to "field", which is very short).
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Martin BrandenburgFeb 10 '12 at 15:19

So the upshot is that I am open for neologisms for the commutative case.
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Martin BrandenburgFeb 10 '12 at 15:21

Isn't this just a bit clumsy, or rather a characterization? Namely, the functor $A \mapsto A^*$ establishes an isomorphism of categories between "field monoids" and abelian groups.
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Martin BrandenburgFeb 9 '12 at 22:44

(Edited to include the word "abelian").
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James CranchFeb 9 '12 at 23:01

I agree it's clumsy, but I'm not wholly convinced you should disguise the characterisation. "Abelian groups with zero" is not all that long compared to some phrases in mathematics.
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James CranchFeb 9 '12 at 23:02

3

Of course, this functor is not quite an equivalence of categories, unless you demand that homomorphisms preserve zero. Otherwise, for example, there's an extra map between any two objects which sends everything (including the zero) to the identity.
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James CranchFeb 9 '12 at 23:02

I don't understand why such comments still arise today (and get upvotes), decades after universal algebra etc. has been established. Of course a homomorphism between monoids with zero is defined to be a map which preserves the whole structure, in particular the zero. There is no reason to apply the forgetful functor to monoids ...
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Martin BrandenburgFeb 10 '12 at 15:13